The molecule of which gas have highest speed ?

Options

A. `H_2 " at " - 73^@C`
B. `CH_4 " at " 300K`
C. `N_2 " at " 1027^@C`
D. `O_2 " at " 0^@C`

Answer

To determine which gas has the highest speed of its molecules, we can use the formula for the root mean square speed of gas molecules, which is given by:

\[ V = \sqrt{\frac{3RT}{M}} \]

Where:
- \( V \) is the root mean square speed,
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas in kg/mol.

### Step-by-Step Solution:

1. **Identify the Gases and Their Molar Masses**:
- Hydrogen (H₂): Molar mass = 2 g/mol = 0.002 kg/mol
- Methane (CH₄): Molar mass = 16 g/mol = 0.016 kg/mol
- Nitrogen (N₂): Molar mass = 28 g/mol = 0.028 kg/mol
- Oxygen (O₂): Molar mass = 32 g/mol = 0.032 kg/mol

2. **Convert Temperatures to Kelvin**:
- For H₂: Assume temperature = 300 K
- For CH₄: Temperature = 300 K
- For N₂: Temperature = 1300 K (1027 °C + 273)
- For O₂: Temperature = 273 K (0 °C)

3. **Calculate the Speed for Each Gas**:
- **For H₂**:
\[ V_{H2} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3 \times R \times 300}{0.002}} \quad \text{(R is a constant)} \]
Since we are comparing, we can ignore the constant \( R \) and focus on the ratio:
\[ V_{H2} \propto \sqrt{\frac{300}{0.002}} = \sqrt{150000} \approx 387.3 \]

- **For CH₄**:
\[ V_{CH4} \propto \sqrt{\frac{300}{0.016}} = \sqrt{18750} \approx 136.9 \]

- **For N₂**:
\[ V_{N2} \propto \sqrt{\frac{1300}{0.028}} = \sqrt{46428.57} \approx 215.4 \]

- **For O₂**:
\[ V_{O2} \propto \sqrt{\frac{273}{0.032}} = \sqrt{8531.25} \approx 92.5 \]

4. **Compare the Speeds**:
- \( V_{H2} \approx 387.3 \)
- \( V_{CH4} \approx 136.9 \)
- \( V_{N2} \approx 215.4 \)
- \( V_{O2} \approx 92.5 \)

5. **Conclusion**:
The gas with the highest speed is Hydrogen (H₂).

### Final Answer:
**The molecule of Hydrogen gas (H₂) has the highest speed.**

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